We present a novel application of Tensor Network methods in cancer treatment as a potential tool to solve the dose optimization problem in radiotherapy. In particular, the intensity-modulated radiation therapy technique-that allows treating irregular and inhomogeneous tumors while reducing the radiation toxicity on healthy organs-is based on the optimization problem of the beamlets intensities that shall result in a maximal delivery of the therapy dose to cancer while avoiding the organs at risk of being damaged by the radiation. The resulting optimization problem is expressed as a cost function to be optimized. Here, we map the cost function into an Ising-like Hamiltonian, describing a system of long-range interacting qubits. Finally, we solve the dose optimization problem by finding the ground-state of the Hamiltonian using a Tree Tensor Network algorithm. In particular, we present an anatomical scenario exemplifying a prostate cancer treatment. A similar approach can be applied to future hybrid classical-quantum algorithms, paving the way for the use of quantum technologies in future medical treatments.
We present a novel application of Tensor Network methods in cancer treatment as a potential tool to solve the dose optimization problem in radiotherapy. In particular, the intensity-modulated radiation therapy technique-that allows treating irregular and inhomogeneous tumors while reducing the radiation toxicity on healthy organs-is based on the optimization problem of the beamlets intensities that shall result in a maximal delivery of the therapy dose to cancer while avoiding the organs at risk of being damaged by the radiation. The resulting optimization problem is expressed as a cost function to be optimized. Here, we map the cost function into an Ising-like Hamiltonian, describing a system of long-range interacting qubits. Finally, we solve the dose optimization problem by finding the ground-state of the Hamiltonian using a Tree Tensor Network algorithm. In particular, we present an anatomical scenario exemplifying a prostate cancer treatment. A similar approach can be applied to future hybrid classical-quantum algorithms, paving the way for the use of quantum technologies in future medical treatments.
The very first successful application of tensor network methods (TNMs) to the solution of the dose optimization in Intensity Modulated Radiation Therapy (IMRT) is considered. This technique provides a method to modulate the local beam intensities, dividing the beam into smaller beamlets. This allows to reduce the radiation toxicity for healthy organs and deal with irregular and inhomogeneous tumors. Plan's goals are encoded as mathematical constraints into a cost function expressing the distance between the prescribed and delivered dose. Aim of the optimization is to minimize the function associating to each beamlet the optimal weight xj. In this thesis a classical quadratic cost function is mapped into an Insing-like Hamiltonian, where the beamlets weights are described by a system of long-range interacting qubits. The aim of the work is to solve the dose optimization problem using a binary tree tensor network (bTTN) to find the Hamiltonian's groundstate and to show the applicability of TNM to the IMRT dose optimization problem.
We apply a new hardware and software platform called the Hamiltonian Engine for Radiotherapy Optimization (HERO) to the problem of Intensity-Modulated Radiation Therapy (IMRT) treatment planning. HERO solves large general-form binary optimization problems by decomposing them into sub-problems and approximating them using a quadratic pseudo-boolean function. Optimizing the resulting function becomes a quadratic unconstrained binary optimization (QUBO) problem, which has been widely studied and has numerous applications in various fields. A Quantum Annealer (QA) approach has been previously investigated to solve QUBO problems, including IMRT optimization. However, the QA can only accommodate a small number of variables and requires several hours to obtain optimized plans. HERO acts as an optimizer for QUBO problems, which not only addresses these shortcomings but also relies solely on conventional hardware design while operating at room temperature. We evaluate HERO on seven prostate IMRT cases with clinical objectives, each using approximately 6000 beamlets. Our method was compared to the commercial treatment planning software, Eclipse, for both time-to-solution and plan quality. HERO solves most cases in about 30 seconds, with significantly lower objective function scores than Eclipse. The results indicate that HERO is promising for radiation therapy optimization problems. Additionally, HERO has the potential to be applied to Volumetric-Modulated Arc Therapy (VMAT) and other complex types of treatment planning.
Optimization methods are critical to radiation therapy. A new technology, quantum annealing (QA), employs novel hardware and software techniques to address various discrete optimization problems in many fields. We report on the first application of quantum annealing to the process of beamlet intensity optimization for IMRT. We apply recently-developed hardware which natively exploits quantum mechanical effects for improved optimization. The new algorithm, called QA, is most similar to simulated annealing, but relies on natural processes to directly minimize a system's free energy. A simple quantum system is slowly evolved into a classical system representing the objective function. If the evolution is sufficiently slow, there are probabilistic guarantees that a global minimum will be located. To apply QA to IMRT-type optimization, two prostate cases were considered. A reduced number of beamlets were employed, due to the current QA hardware limitations. The beamlet dose matrices were computed using CERR and an objective function was defined based on typical clinical constraints, including dose-volume objectives, which result in a complex non-convex search space. The objective function was discretized and the QA method was compared to two standard optimization methods, simulated annealing and Tabu search, run on a conventional computing cluster. Based on several runs, the average final objective function value achieved by the QA was 16.9 for the first patient, compared with 10.0 for Tabu and 6.7 for the simulated annealing (SA) method. For the second patient, the values were 70.7 for the QA, 120.0 for Tabu and 22.9 for the SA. The QA algorithm required 27-38% of the time required by the other two methods. In this first application of hardware-enabled QA to IMRT optimization, its performance is comparable to Tabu search, but less effective than the SA in terms of final objective function values. However, its speed was 3-4 times faster than the other two methods. This initial experiment suggests that more research into QA-based heuristics may offer significant speedup over conventional clinical optimization methods, as quantum annealing hardware scales to larger sizes.
Hybrid tensor networks (hTNs) offer a promising solution for encoding variational quantum states beyond the capabilities of efficient classical methods or noisy quantum computers alone. However, their practical usefulness and many operational aspects of hTN-based algorithms, like the optimization of hTNs, the generalization of standard contraction rules to an hybrid setting, and the design of application-oriented architectures have not been thoroughly investigated yet. In this work, we introduce a novel algorithm to perform ground-state optimizations with hybrid tree tensor networks (hTTNs), discussing its advantages and roadblocks, and identifying a set of promising applications. We benchmark our approach on two paradigmatic models, namely the Ising model at the critical point and the Toric-code Hamiltonian. In both cases, we successfully demonstrate that hTTNs can improve upon classical equivalents with equal bond dimension in the classical part.
We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.
Quantum computing has long promised to revolutionize the way we solve complex problems. At the same time, tensor networks are widely used across various fields due to their computational efficiency and capacity to represent intricate systems. While both technologies can address similar problems, the primary aim of this thesis is not to compare them. Such comparison would be unfair, as quantum devices are still in an early stage, whereas tensor network algorithms represent the state-of-the-art in quantum simulation. Instead, we explore a potential synergy between these technologies, focusing on how two flagship algorithms from each paradigm, the Density Matrix Renormalization Group (DMRG) and quantum annealing, might collaborate in the future. Furthermore, a significant challenge in the DMRG algorithm is identifying an appropriate tensor network representation for the quantum system under study. The representation commonly used is called Matrix Product Operator (MPO), and it is notoriously difficult to obtain for certain systems. This thesis outlines an approach to this problem using finite automata, which we apply to construct the MPO for our case study. Finally, we present a practical application of this framework through the quadratic knapsack problem (QKP). Despite its apparent simplicity, the QKP is a fundamental problem in computer science with numerous practical applications. In addition to quantum annealing and the DMRG algorithm, we implement a dynamic programming approach to evaluate the quality of our results. Our results highlight the power of tensor networks and the potential of quantum annealing for solving optimization problems. Moreover, this thesis is designed to be self-explanatory, ensuring that readers with a solid mathematical background can fully understand the content without prior knowledge of quantum mechanics.